I have a part with some oddly shaped pads specified in the part's recommended footprint. It seems that in the footprint editor, there are only 3 options for pin shapes. Therefore, I chose the closest one (square pad) and then extended them to match the recommended footprint using filled polygons in the metal layer.

My problem is that I get DRC errors indicating that the polygons are too close to the pins. I cannot find a way to tell the footprint editor that the polygons share the same net as the pins. Normally on the general layout, I can specify to which net the polygon belongs, however in footprint editor, this menu is grayed-out.

Once you have created a pad you cannot change the net for it. You will continue to get this error. But since you know that it will actually work I would recommend you simply ignore that error.You may however run into problems in maufacturing since I don't think the custom pins will not show up in the Soldermask layer. What you'll need to do is manually input the pins onto the Soldermask layer in the exact same location to ensure that they are not plated over. To do this create a polygon around the current pin that you have, then copy that polygon and paste it onto the Soldermask layer. Then repeat that for each pin in their locations. That should solve any problems you have.